Problem: Simplify and expand the following expression: $ \dfrac{5y + 10}{3y - 5}+\dfrac{3y - 2}{y - 8} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(3y - 5)(y - 8)$ Multiply the first term by $\dfrac{y - 8}{y - 8}$ $ \begin{align*} \dfrac{5y + 10}{3y - 5} \times \dfrac{y - 8}{y - 8} & = \dfrac{(5y + 10)(y - 8)}{(3y - 5)(y - 8)} \\ & = \dfrac{5y^2 - 30y - 80}{(3y - 5)(y - 8)}\end{align*} $ Multiply the second term by $\dfrac{3y - 5}{3y - 5}$ $ \begin{align*} \dfrac{3y - 2}{y - 8} \times \dfrac{3y - 5}{3y - 5} & = \dfrac{(3y - 2)(3y - 5)}{(y - 8)(3y - 5)} \\ & = \dfrac{9y^2 - 21y + 10}{(y - 8)(3y - 5)}\end{align*} $ Now we have: $ = \dfrac{5y^2 - 30y - 80}{(3y - 5)(y - 8)} + \dfrac{9y^2 - 21y + 10}{(y - 8)(3y - 5)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{5y^2 - 30y - 80 + 9y^2 - 21y + 10}{(3y - 5)(y - 8)} $ $ = \dfrac{14y^2 - 51y - 70}{(3y - 5)(y - 8)}$ Expand the denominator: $ = \dfrac{14y^2 - 51y - 70}{3y^2 - 29y + 40}$